Normalize the <cubic anomaly coefficient> of the $SU(N)$ fundamental to $+1$. The <two-index antisymmetric representation> has anomaly coefficient $N-4$, while an antifundamental contributes $-1$. Cancellation of the $SU(N)^3$ <gauge anomaly> requires
$$
(N-4)-p=0,
\qquad
\boxed{p=N-4}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 307 / 3 / i / Solution

The electric quarks give $2N_c$ copies of the $SU(2N_f)$ fundamental, so
$$
\mathcal A_{\rm el}[SU(2N_f)^3]=2N_c.
$$
In the magnetic theory the $2\widetilde N_c$ color components of $q$ transform as flavor antifundamentals and contribute $-2\widetilde N_c$. The singlet $M^{ij}$ is the <two-index antisymmetric representation> of $SU(2N_f)$, whose <cubic anomaly coefficient> is $2N_f-4$. Hence
$$
\begin{aligned}
\mathcal A_{\rm mag}[SU(2N_f)^3]
&=-2(N_f-N_c-2)+(2N_f-4)\\
&=\boxed{2N_c}
=\mathcal A_{\rm el}.
\end{aligned}
$$
This is a direct anomaly check of <Seiberg duality>.

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 307 / 4 / viii / Solution

{title2=$\Lambda^2V$}

The two-index antisymmetric representation of $SU(N)$ is the exterior square of its fundamental representation. It has dimension $N(N-1)/2$, Dynkin index $N-2$, and cubic anomaly coefficient $N-4$ in the same normalization.


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 Incoming links: Two-index antisymmetric representation